Obtaining cyclic redundancy code

ABSTRACT

Cyclic redundancy codes are obtained to verify the integrity of a message transmitted between a sender and a receiver. One method for obtaining a cyclic redundancy code includes separating the message into segments. Remainders are obtained for those segments based on a generator polynomial. The remainders for those segments are multiplied by a segment-constant to obtain segment-remainders for each segment. The segment-remainders are accumulated into an accumulated-remainder. The accumulated-remainder is moduloed by the generator polynomial to obtain a remainder for the accumulated-remainder. The remainder for the accumulated-remainder is the cyclic redundancy code for the message.

TECHNICAL FIELD

[0001] This application relates to verifying the integrity of data transmissions, and more particularly, to verifying the integrity of digital data transmissions using cyclic redundancy codes.

BACKGROUND

[0002] Data transmission 10 (FIG. 1) involves a transfer of information known as data between a sender 12 and a receiver 14. Often, data transmission 10 includes information transmitted as digital bits of ones and zeros, represented here by m_(n−1) to m₀, and referred to as a message M.

[0003] In a perfect world, message M transmits free of errors. Unfortunately, errors are often introduced though medium 16 by which message M travels from sender 12 to receiver 14 (e.g., medium 16 may be any combination of wire, cable, fiber-optic, air, or link layer devices). One method for detecting the presence of errors in message M employs cyclic redundancy codes.

[0004] Cyclic redundancy codes treat groupings of digital bits like message M as a polynomial where each bit in the grouping represents a coefficient in the polynomial X^(n−1)+X^(n−2)+X⁰. For example, a group of eight bits 11001101 may be represented by polynomial X⁷+X⁶+X³+X²+1 (i.e., 1*X⁷+1*X⁶+0*X⁵+0*X⁴ +1*X ³+1*X² +0*X ¹+1*X⁰).

[0005] These polynomials form an algebraic object known as a commutative ring with coefficients in Z/p where Z are the integers and p is a prime number, here 2, also known as {0, 1} modulo 2. A non empty set R together with two binary operations {+*} is called a ring if (R+) is an abelian group, (R*) is a semi group and the distributive laws are obeyed, (i.e. a*(b+c)=a*b+a*b).

[0006] In polynomial rings, there are no carries or borrows from one coefficient to the next. In arithmetic modulo 2, addition and subtraction are identical and may be implemented with exclusive-or. $\begin{matrix} 10111011 & 10110011 & 11110000 & 00010101 \\ \underset{\_}{{+ 11001110}\quad} & \underset{\_}{{+ 11000101}\quad} & \underset{\_}{{- 00100110}\quad} & {\underset{\_}{- 10101111}\quad} \\ 01110101 & 01110110 & 11010110 & 10111010 \end{matrix}$

[0007] Division of polynomials represented as groups of bits is completed in a manner similar to binary division except subtraction is done in modulo2. A divisor will ‘go into’ a dividend if the dividend polynomial is of the same degree as the divisor polynomial (i.e., the divisor and dividend share at least the same most significant bit).

[0008] A cyclic redundancy code may be obtained by calculating the remainder for message M divided by a generator polynomial P. This remainder is called a cyclic redundancy code (“CRC”).

[0009] To obtain a CRC for a message M, the group of bits to be divided by generator polynomial P may include appended zero-bits 17. Zero-bits 17 are equal in number to the degree of generator polynomial P. Thus, the CRC of a message M=10111000 having three appended zero-bits 17 based on a generator polynomial P=X³+1=1001 of degree three (i.e., where X³ is the most significant bit of polynomial P) may be calculated as follows:

[0010] The resulting remainder, shown as CRC 18, may be appended to message M, replacing zero bits 17, to create a message M′. Sender 12 transmits message M′ via medium 16 to receiver 14 as data transmission 10.

[0011] Upon receipt, receiver 14 divides message M′ by the same generator polynomial P to obtain a CRC for M′ and check the validity of data transmission 10. If the resulting remainder is zero (i.e., CRC=M′(modulo)P=0), the integrity of data transmission 10 is confirmed. For example:

[0012] If the remainder of message M′ divided by polynomial P is not zero (i.e., CRC=M′(modulo)P≠0), data transmission 10 contains one or more errors. For example:

DESCRIPTION OF DRAWINGS

[0013]FIG. 1 shows a prior art block diagram of a data transmission employing cyclic redundancy codes.

[0014]FIG. 2 shows a process for obtaining a CRC where a message M is separated into a plurality of segments.

[0015]FIG. 3 shows a CRC generator for obtaining a CRC of a message according to the process in FIG. 2.

[0016]FIG. 4 shows the CRC generator of FIG. 3 separated into four segments.

[0017]FIG. 5 shows a CRC generator for obtaining a CRC of a message M according to the process in FIG. 2.

[0018]FIG. 6 shows a modulo unit for obtaining a remainder of a message using a reciprocal approximation of a generator polynomial.

[0019]FIG. 7 shows the CRC generator in FIG. 3 using the modulo unit in FIG. 6.

[0020]FIG. 8 shows a CRC generator for updating a CRC of a message M where a portion of message M has been adjusted.

[0021]FIG. 9 shows the CRC generator in FIG. 8 operating on an adjusted segment of message M.

[0022]FIG. 10 is a view of computer hardware used to implement an embodiment of the invention.

[0023] Like reference symbols in the various drawings indicate like elements.

DETAILED DESCRIPTION

[0024] Process 20 (FIG. 2) obtains a CRC for a message M based on a generator polynomial P.

[0025] Process 20 includes separating (201) a message M into a plurality of message segments M_(s); moduloing (defined below) (203) the message segments by a generator polynomial P, if needed, to obtain a remainder R for each segment; multiplying (205) the remainder R for each segment by an appropriate segment-constant C to obtain a segment-remainder SR for each segment M_(s); accumulating (207) the segment-remainders SR for each segment M_(s) to obtain an accumulated-remainder AR for message M; and moduloing (209) the accumulated-remainder by generator polynomial P, if needed, to obtain the CRC for message M.

[0026] Separating (201) message M into a plurality of message segments M_(s) includes parsing message M so that:

M=M _(s−1) *X ^(n*(s−1))+M _(s−2) *X ^(n*(s−2)) . . . +M ₁ *X ^(n(1)) +M ₀ *X ^(n(0));

[0027] where s is the number of segments into which message M is separated, n is the number of bits in each segment, X is the position of each segment in message M, and M_(s) are the individual segments of message M. If the number of bits is n, then X is of the form X=[1000 . . . 0] where there are n zeroes, n+1 elements, and X is of degree n. Multiplying M by X will shift the message left by n bits. Multiplying M by X² will shift the message by 2n bits (and so on).

[0028] Moduloing (203) includes obtaining a remainder R for each message segment M_(s) by dividing segment M_(s) by generator polynomial P if the degree of the most significant bit of segment M_(s) is the same as or greater than the degree of the most significant bit of polynomial P. If the degree of segment M_(s) is less than the degree of polynomial P (i.e., where the most significant bit of M_(s) is smaller than the most significant bit of polynomial P) moduloing (203) is not needed since the remainder for message segment M_(s) equals segment M_(s) itself. In alternate embodiments moduloing (203) may be accomplished by multiplying message segment M_(s) by a reciprocal approximation for polynomial P, rather than dividing segment M_(s) by polynomial P. to obtain remainder R for message segment M_(s). The operation of multiplication by reciprocal approximation to obtain a remainder R is discussed in connection with FIG. 6 below.

[0029] Multiplying (205) includes obtaining segment-constant C (defined below) for each message segment M_(s) and multiplying each segment-constant C by its remainder R to obtain a segment-remainder SR for each message segment. Segment-constants C may be obtained based on the position X of message segment M_(s) in message M modulo generator polynomial P or modulo a field extension of P.

[0030] Accumulation (207) includes adding the segment-remainders SR for each message segment M_(s) to obtain an accumulated-remainder AR for message M.

[0031] Moduloing (209) includes dividing accumulated-remainder AR by generator polynomial P, or multiplying AR by a reciprocal approximation of generator polynomial P, to obtain a CRC for message M. However, if the degree of accumulated-remainder AR is less than the degree of polynomial P, moduloing (209) is not needed since the remainder (i.e., the CRC) of message M is accumulated-remainder AR.

[0032]FIG. 3 shows an implementation of process 20 for calculating a CRC of message M based on generator polynomial P. For example:

[0033] if

[0034] M=10111000, (Message 10111 with with 3 zero-bits appended)

[0035] s=2,

[0036] n=4, and

[0037] P=1001=(Deg(P)=3);

[0038] then M may be separated as

[0039] M_(s−1)=M₁=1011=segment 33,

[0040] X^(n*(s−1))=X⁴=10000,

[0041] M_(s−2)=M₀=1000=segment 35,

[0042] X^(n*(s−2))=X⁰=00001;

[0043] where

[0044] M=1011*10000+1000*00001=10111000.

[0045] CRC generator 30 obtains a CRC for message M based on generator polynomial P, where the CRC for message M is the remainder of message M divided by polynomial P (i.e., CRC=M(modulo)P=MOD(M, P).

[0046] Typically, generator polynomials P are selected because they are irreducible (i.e., they have no factors). Several examples of well-known generator polynomials include: $\begin{matrix} {{LRCC8} = {X^{8} + 1}} \\ {{= 100000001};} \\ {{CRC16} = {x^{16} + x^{15} + X^{2} + 1}} \\ {{= {110000000000000\quad 101}};} \\ {{SDLC} = {x^{16} + x^{12} + X^{5} + 1}} \\ {{= {1000100000010000\quad 1}};} \\ {{LRCC} = {x^{16} + 1}} \\ {{= {1000000000000000\quad 1}};} \\ {{CRC12} = {x^{12} + x^{11} + X^{3} + X^{2} + X + 1}} \\ {{= 1100000001111}\quad;{and}} \\ {{ETHERNET} = {x^{32} + x^{26} + X^{23} + X^{22} + x^{16} + x^{11} + X^{10} + X^{8} + x^{7} + X^{5} + X^{4} + X^{2} + X + 1}} \\ {{= {100000100\quad 110000010000\quad 110\quad 110110111}};} \end{matrix}$

[0047] where LRCC stands for Last Registration Control Channel.

[0048] CRC generator 30 includes modulo unit 32, multiplier 34, accumulator 36, and modulo unit 38. Here, modulo unit 32 has modulo units 32 a and 32 b implemented in hardware.

[0049] Modulo unit 32 a divides message segment 33 by generator polynomial P to obtain remainder R_(i+1) (i.e., R_(i+1)=M_(s−1)(modulo)P=MOD(M_(s−1), P)). Modulo unit 32 b divides message segment 35 by generator polynomial P to obtain remainder R_(i) (i.e., R_(i)=M_(s−2)(modulo)P=MOD(M_(s−2), P)). For example:

[0050] if

[0051] M=10111000,

[0052] M_(s−1)=M₁=1011=segment 33,

[0053] M_(s−2)=M₀=1000=segment 35, and

[0054] P=1001;

[0055] then

[0056] R_(i+1)=R₁=M_(s−1)(modulo)P=MOD(M_(s−1), P)

[0057] =1011(modulo)1001=010, and

[0058] R_(i)=R₀=M_(s−2)(modulo)P=MOD(M_(s−2), P)

[0059] =1000(modulo)1001=001;

[0060] where

[0061] Multiplier 34 multiplies remainders R_(i+1) and R_(i) by segment-constants C_(i+1) and C_(i) to obtain segment-remainders SR_(i+1) and SR_(i). Here, segment-constants C_(i+1) and C_(i) are obtained by moduloing the position X of segments 33 and 35 in message M by generator polynomial P (i.e., C_(i+1)=X^(n* (i+1))(modulo)P and C_(i)=X^(n*i)(modulo)P). For example:

[0062] if

[0063] M=10111000,

[0064] P=1001,

[0065] s=2, and

[0066] n=4;

[0067] then for

[0068] SR_(i+1)=SR₁

[0069] C_(i+1)=C₁=X^(4*1)(modulo)P

[0070] =10000(modulo)1001=010, and

[0071] SR_(i)=SR₀

[0072] C_(i)=C₀=X^(4*0)(modulo)P

[0073] =0001(modulo)1001=001;

[0074] where

[0075] Segment-constants C_(i+1) and C_(i) may be obtained in advance, based on a known segmentation of message M and stored in memory unit 39, which is accessible to CRC generator 30. In other embodiments, segment-constants C_(i+1) and C_(i) may be obtained ‘on the fly’ within CRC generator 30 upon receipt of message M.

[0076] Multiplier 34 includes multipliers 34 a and 34 b. Multiplier 34 a multiplies remainder R_(i+1) by segment-constant C_(i+1) to obtain segment-remainder SR_(i+1). Multiplier 34 b multiplies remainder R_(i) by segment constant C_(i) to obtain segment-remainder SR_(i). For example:

[0077] if

[0078] R_(i+1)=010, C_(i+1)=010 and (from above)

[0079] R_(i)=001, C_(i)=001;

[0080] then

[0081] SR_(i+1)=R_(i+1)*C_(i+1)=010*010=00100, and

[0082] SR_(i)=R_(i)*C₁=001*001=00001;

[0083] where $\begin{matrix} 010 & 001 \\ \underset{\_}{{*\quad 010}\quad} & \underset{\_}{{*\quad 001}\quad} \\ {{+ 000}\quad} & {{+ 001}\quad} \\ {{+ 010}\quad} & {{+ 000}\quad} \\ {\underset{\_}{+ 000}\quad} & {\underset{\_}{+ 000}\quad} \\ {\quad {00100\quad = {SR}_{i + 1}}\quad} & {\quad {00001\quad = {{SR}_{i}.}}\quad} \end{matrix}$

[0084] Accumulator 36 adds segment-remainders SR_(i+1) and SR_(i) together to obtain accumulated-remainder AR. For example:

[0085] if

[0086] SR_(i+1)=SR₁=00100, (from above)

[0087] SR_(i)=SR₀=00001;

[0088] then

[0089] AR=00100+00001=00101,

[0090] where $\begin{matrix} {00100\quad} \\ {\underset{\_}{{+ 00001}\quad}\quad} \\ {00101 = {{AR}.}} \end{matrix}$

[0091] Modulo unit 38 obtains the CRC for message M by moduloing accumulated-remainder AR by generator polynomial P (i.e., CRC=AR(modulo)P=MOD(AR, P)). For example:

[0092] if

[0093] AR=00101, and (from above)

[0094] P=1001;

[0095] then

[0096] CRC=AR(modulo)P=MOD(AR, P)

[0097] =00101(modulo)1001=101,

[0098] where

[0099] Hence, process 20 implemented on CRC generator 30 obtains the same CRC for message M, here 10111000. In this example, moduloing AR by polynomial P is not needed since the degree of AR was less than the degree of P.

[0100] CRC generator 30 may be expanded to include enough components for obtaining the CRC for message M separated into N segments. FIG. 4 shows CRC generator 40 capable of operating on message M separated into four (4) segments 43, 45, 47 and 49. For example:

[0101] if

[0102] M=M′=10111101, (e.g., message M in the example for FIG. 3 above having the obtained CRC appended to it)

[0103] s=4,

[0104] n=2, and

[0105] p=2;

[0106] then M may be separated as

[0107] M_(s-1)=M₃=10=segment 43,

[0108] X^(n*(s-1))=X⁶=1000000,

[0109] M_(s-2)=M₂=11=segment 45,

[0110] X^(n*(s-2))=X⁴=10000,

[0111] M_(s-3)=M₁=11=segment 47,

[0112] X^(n*(s-3))=X²=100,

[0113] M_(s-4)=M₀=01=segment 49,

[0114] X^(n*(s-4))=X⁰=001;

[0115] where

[0116] M=10*1000000+11*10000+11*100+01*001=10111101.

[0117] CRC generator 40 includes modulo unit 42, multipliers 44, accumulator 46, and modulo unit 48. Modulo unit 42 includes modulo units 42 a, 42 b, 42 c and 42 d. Modulo units 42 a, 42 b, 42 c and 42 d each operate to divide message segment 43, 45, 47 and 49 by generator polynomial P to obtain remainders R₃, R₂, R₁ and R₀. For example:

[0118] if

[0119] M=10111101=M′, (from above)

[0120] M_(s−1)=M₃=10=segment 43,

[0121] M_(s−2)=M₂=11=segment 45,

[0122] M_(s−3)=M₁=11=segment 47,

[0123] M_(s−4)=M₀=01=segment 49, and

[0124] P=1001;

[0125] then

[0126] R_(i+3)=R₃=M_(s−1)(modulo)P=10(modulo)1001

[0127] =10,

[0128] R_(i+2)=R₂=M_(s−2)(modulo)P=11(modulo)1001

[0129] =11,

[0130] R₁₊₁=R₁=M_(s−3)(modulo)P=11(modulo)1001

[0131] =11, and

[0132] R_(i)=R₀=M_(s−4)(modulo)P=01(modulo)1001

[0133] =01.

[0134] Multiplier 44 multiplies remainders R₃ to R₀ by segment-constants C₃ to C₀ to obtain segment-remainders SR₃ to SR₀. Segment-constants C₃ to C₀ correspond to each particular segment 43, 45, 47 and 49 and may be obtained by moduloing the position of segments in message M by polynomial P. (i.e., C₃=X^(n*(3))(modulo)P, C₂=X^(n*2)(modulo)P, C₁=X^(n*1)(modulo)P, C₀=X^(n*0)(modulo)P). For example:

[0135] if

[0136] M=10111101, (from above)

[0137] P=1001,

[0138] s=4, and

[0139] n=2;

[0140] then

[0141] SR₃=SR_(i+3)

[0142] C₃=C_(i+3)=X^(2*3)(modulo)P

[0143] =1000000(modulo)1001=001;

[0144] SR₂=SR₀

[0145] C₂=C_(i+2)=X^(2*2)(modulo)P

[0146] =10000(modulo)1001=010;

[0147] SR₁=SR₁

[0148] C₁=C_(i+1)=X^(2*1)(modulo)P

[0149] 100(modulo)1001=100; and

[0150] SR₀=SR_(i)

[0151] C₀=C_(i+0)=X^(4*0)(modulo)P

[0152] =001(modulo)1001=001.

[0153] Segment constants C₃ to C₀ may be obtained in advance based on the segmentation of message M and stored in a memory unit 39 (FIG. 3) accessible to CRC generator 40. In other embodiments C₃ to C₀ may be obtained ‘on the fly’ (i.e., in real-time)within CRC generator 40 as it receives message M.

[0154] Multiplier 44 multiplies R₃ by C₃, R₂ by C₂, R₁ by C₁, and R₀ by C₀ to obtain segment-remainders SR₃ to SR₀. For example:

[0155] if

[0156] R_(i+3)=R₃=10, C_(i+3)=C₃=001; (from above)

[0157] R_(i+2)=R₂=11, C_(i+2)=C₂=010;

[0158] R_(i+1)=R₁=11, C_(i+1)=C₁=100; and

[0159] R_(i=R) ₀=01, C_(i)=C₁₀=001;

[0160] then

[0161] SR₃=R₃*C₃=10*001=0010;

[0162] SR₂=R₂*C₂=11*010=0110;

[0163] SR₁=R₁*C₁=11*100=1100; and

[0164] SR₀=R₀*C₀=01*001=0001.

[0165] Accumulator 46 adds segment-remainders SR₃ to SR₀ together to obtain accumulated-remainder AR. Here, accumulator 46 includes accumulators 46 a, 46 b and 46 c, where accumulators 46 a and 46 b compute temporary accumulations T₁ and T₀ and accumulator 46 c combines temporary accumulations T₁ and T₀ to obtain accumulated-remainder AR. For example:

[0166] if

[0167] SR_(i+3)=SR₃=0010, (from above)

[0168] SR_(i+2)=SR₂=0110,

[0169] SR_(i+1)=SR₁=1100, and

[0170] SR_(i)=SR₀=0001;

[0171] then

[0172] T₁=0010+0110=0100,

[0173] T₀=1100+0001=1101, and

[0174] AR=0100+1101=1001.

[0175] Finally, modulo unit 48 obtains the CRC for message M, here message M′ having the CRC obtained as described in FIG. 3 above, by moduloing accumulated-remainder AR by polynomial P (i.e., CRC=AR(modulo)P=MOD(AR, P). For example:

[0176] if

[0177] AR=1001, and (from above)

[0178] P=1001;

[0179] then

[0180] CRC=AR(modulo)P=1001(modulo)1001

[0181] =0

[0182] where

[0183] Thus, CRC generator 40 verifies the integrity of message M from the example in FIG. 3 where the CRC of message M was appended to form M′ and transmitted to a receiver 14 who confirmed the transmission using CRC generator 40 (FIG. 4).

[0184] According to process 20, CRC generators 30 and 40 may be further simplified where the degree of message segments M_(s) are less than the degree of generator polynomial P (i.e., Deg(M_(s))<Deg(P)). As shown in the example for FIG. 4 above, the remainder R of M_(s)(modulo)P equals M_(s) when the degree of M_(s) is less than the degree of P. Thus, CRC generator 50 (FIG. 5) does not need an initial modulo unit (e.g., 32 or 42) for obtaining a remainder R_(i) of message segments M_(s) that are of a degree less than the degree of generator polynomial P. For segments of degree equal to P (i.e., Deg(M_(s))=Deg(P)) modulo units 32 or 42 may be replaced by an xor, as M_(s)(modulo)P equals M_(s)−P.

[0185] Here, CRC generator 50 includes multiplier 54, accumulator 56, and modulo unit 58, which operate to obtain a CRC for message M separated into four segments 53, 55, 57 and 59 of a degree less than the degree of generator polynomial P (i.e., Deg(M_(s))<Deg(P)). For example:

[0186] if

[0187] M=10111000, (M including 3 appended zero bits as in FIG. 3 above)

[0188] s=4,

[0189] n=2, and

[0190] P=1001;

[0191] then

[0192] M_(s-1)=M₃=10=segment 53,

[0193] X^(n*(s-1))=X⁶=1000000,

[0194] M_(s-2)=M₂=11=segment 55,

[0195] X^(n*(s-2))=X⁴=10000,

[0196] M_(s-3)=M₁=10=segment 57,

[0197] X^(n*(s-3))=X²=100,

[0198] M_(s-4)=M₀=00=segment 59,

[0199] X^(n*(s-4))=X⁰=001;

[0200] and M = 10 * 1000000 + 11 * 10000 + 10 * 100 + 00 * 001 = 10111000.

[0201] Multiplier 54 multiplies segments 53 to 59 by segment-constants C₃ to C₀ to obtain segment-remainders SR₃ to SR₀. Segment-constants C₃ to C₀ may be obtained in advance or calculated ‘on the fly’ as described above. For example:

[0202] if

[0203] M=10111000, (from above)

[0204] P=1001,

[0205] s=4, and

[0206] n=2;

[0207] then

[0208] SR₃=SR_(i+3)

[0209] C₃=C_(i+3)=X^(2*3)(modulo)P

[0210] =1000000(modulo)1001=001;

[0211] SR₂=SR_(i+2)

[0212] C₂=C_(i+2)=X^(2*2)(modulo)P

[0213] =10000(modulo)1001=010;

[0214] SR₁=SR_(i+1)

[0215] C₁=C_(i+1)=X^(2*1)(modulo)P

[0216] =100(modulo)1001=100; and

[0217] SR₀=SR_(i)

[0218] C₀=C_(i+0)=X^(4*0)(modulo)P

[0219] =001(modulo)1001=001.

[0220] Multiplier 54 multiplies M₃ by C₃, M₂ by C₂, M₁ by C₁, and M₀ by C₀ to obtain segment-remainders SR₃ to SR₀, since each message segment M_(s) equals its remainder R. For example:

[0221] if

[0222] M_(s−1)=M₃=10, C_(i+3)=C₃=001,

[0223] M_(s−2)=M₂=11, C₁₊₂=C₂=010,

[0224] M_(s−3)=M₁=10, C_(i+1)=C₁=100, and

[0225] M_(s−4)=M₀=00, C_(i=C) _(i0)=001;

[0226] then

[0227] SR₃=M₃*C₃=10*001=0010,

[0228] SR₂=M₂*C₂=11*010=0110,

[0229] SR₁=M₁*C₁=10*100=1000, and

[0230] SR₀=M₀*C₀=00*001=0000.

[0231] Accumulator 56 adds segment-remainders SR₃ to SR₀ together to obtain accumulated-remainder AR. Here, accumulator 56 includes accumulators 56 a, 56 b and 56 c, where accumulators 56 a and 56 b compute temporary accumulations T₁ and T₀ and accumulator 56 c combines temporary accumulations T₁ and T₀ to obtain accumulated-remainder AR. For example:

[0232] if

[0233] SR_(i+3)=SR₃=0010, (from above)

[0234] SR_(i+2)=SR₂=0110,

[0235] SR_(i+1)=SR₁=1000, and

[0236] SR_(i)=SR₀0000;

[0237] then

[0238] T₁=0010+0110=0100;

[0239] T₀=1000+0000=1000; and

[0240] AR=0100+1000=1100.

[0241] Finally, modulo unit 58 obtains a CRC for message M by moduloing accumulated-remainder AR by polynomial P. For example:

[0242] if

[0243] AR=1100, and (from above)

[0244] P=1001;

[0245] then

[0246] CRC=AR(modulo)P=1100(modulo)1001

[0247] =101;

[0248] where

[0249] Thus, CRC generator 50 obtains the same CRC for message M as calculated in the example in FIG. 3 above without needing modulo units 32 or 42 of FIGS. 3 and 4.

[0250] Moduloing (e.g., (203) and (209)) may also be accomplished by multiplying message M (or message segment M_(s)) by a reciprocal approximation D of generator polynomial P and subtracting that result from message M (or message segment M_(s)) to obtain a remainder R. Moduloing by multiplication by reciprocal approximator RA may be obtained based upon the following relationships:

[0251] RA=X^(p+d)/P;

[0252] M/P=M*RA*1/X^(p+ra) (for 0<=Deg(M)<=p+ra);

[0253] M=(M/P)*P+M(modulo)P;

[0254] R=M(modulo)P=M−(M/P)*P;

[0255] R=M(modulo)P=M−(M*D/X^(p+ra))*P

[0256] where X^(p+ra) is a polynomial having a most significant bit of degree p+ra (i.e. Deg(X^(p+ra))=p+ra); p is the degree of generator polynomial P (i.e., Deg(P)=p); ra is the degree of reciprocal-approximator RA (i.e., Deg(RA)=ra); and the degree of message M, for which remainder R is sought, is greater than zero and less than or equal to p+ra (i.e., 0<Deg(M)<=p+ra). For example:

[0257] if

[0258] M=10111000 (i.e., Deg(M)=7), and

[0259] P=1001 (i.e., Deg(P)=3).

[0260] then reciprocal-approximator RA would have a degree of at least four (4) for p+ra to be greater than or equal to the degree of M, here seven (7). Thus:

[0261] if

[0262] M=10111000 (i.e., Deg(M)=7),

[0263] P=1001 (i.e., Deg(P)=3); and;

[0264] ra=4;

[0265] then

[0266] X^(p+ra)=10000000 (i.e., Deg(X^(p+ra))=7), and

[0267] D=X^(p+ra)/P=X³⁺⁴/1001

[0268] =10000000/1001=10010;

[0269] where

[0270] Modulo unit 60 may calculate reciprocal-approximator RA prior to receiving message M and store RA in memory 69 since both generator polynomial P and the degree of message M are known prior to receiving message M. In other embodiments, reciprocal-approximator RA may be built in or obtained ‘on-the fly’ by modulo unit 60 after receiving message M. Once the form of the polynomial is fixed, the implementation of the corresponding hardware may be simplified considerably.

[0271] To obtain remainder R for message M modulo unit 60 includes multiplication unit 62, truncation unit 64, multiplication unit 66 and subtraction unit 68 where:

[0272] T₀=M*RA is performed by unit 62,

[0273] T₁=T₀/X^(p+ra) is performed by unit 64,

[0274] T₂=T₁*P is performed by unit 66, and

[0275] R=M−T₂ is performed by unit 68.

[0276] Multiplication unit 62 receives message M and multiplies M by reciprocal-approximator RA to obtain temporary result T₀. For example:

[0277] if

[0278] M=10111000, (from FIG. 3 above)

[0279] P=1001, and

[0280] RA=10010;

[0281] then

[0282] T₀=M*RA=10111000*10010

[0283]  =101011110000;

[0284] where

[0285] Multiplication unit 62 provides temporary result T₀ to truncation unit 64, which divides T₀ by X^(p+ra), here 10000000, to obtain truncated result T₁. In other embodiments, truncation unit 64 may remove the p+ra least significant bits of temporary result T₀ without dividing by X^(p+ra) to obtain truncated result T₁. For example:

[0286] if

[0287] P=3,

[0288] ra=4, and

[0289] T₀=101011110000;

[0290] then

[0291] p+ra=7, and

[0292] T₁=10101.

[0293] Thus for p+ra equaling seven (7), the seven (7) least significant bits, here 1110000, are removed from T₀ to obtain T₁.

[0294] Truncation unit 64 provides truncated result T₁ to multiplication unit 66, which multiplies T₁ by generator polynomial P to obtain temporary result T₂. For example;

[0295] if

[0296] P=1001, and

[0297] T₁=10101;

[0298] then

[0299] T₂=T₁*P=10101*1001

[0300] =10111101

[0301] where

[0302] Multiplication unit 66 provides temporary result T₂ to subtraction unit 68, which subtracts T₂ from message M to obtain remainder R. For example:

[0303] if

[0304] M=10111000, and (from above)

[0305] T₂=10111101;

[0306] then

[0307] R=M−T₂=101

[0308] where

[0309] Thus, modulo unit 60 obtains remainder R for message M using multiplication by reciprocal approximation. Hence, modulo unit 60 may calculate the CRC for the entire message M on its own, or may be incorporated into CRC generators 30 and 40 to obtain remainders R for message segments M_(s).

[0310] For example, FIG. 7 shows an implementation of the CRC generator in FIG. 3 employing modulo unit 60 in FIG. 6. Here, modulo units 60 are show as MH (M, RA, P). For example:

[0311] if

[0312] M=10111000, (Same as in FIG. 3 above)

[0313] s=2,

[0314] n=4, and

[0315] P=1001;

[0316] then M may be separated as

[0317] M_(s−1)=M₁=1011=segment 73

[0318] X^(n*(s−1))=X⁴=10000,

[0319] M_(s−2)=M₀=1000=segment 75

[0320] X^(n*(s−2))=X⁰=00001;

[0321] where

[0322] M=1011*10000+1000*00001=10111000.

[0323] CRC generator 70 obtains a CRC for message M based on generator polynomial P, where the CRC for message M is the remainder of message M divided by polynomial P.

[0324] CRC generator 70 includes modulo unit 72, multiplier 74, accumulator 76, and modulo unit 78. Here, modulo unit 72 includes modulo units 72 a and 72 b, which multiply message segments 73 and 75 by a reciprocal approximation of generator polynomial P to obtain remainders R_(i+1) and R_(i).

[0325] Modulo unit 72 a multiplies message segment 73 by reciprocal-approximator RA of generator polynomial P to obtain a remainder R as shown in FIG. 6. For example:

[0326] if

[0327] M=10111000,

[0328] M_(s−1)=M₁=1011=segment 73,

[0329] M_(s−2)=M₀=1000=segment 75,

[0330] Deg(M_(s−1))=3

[0331] Deg(M_(s−2))=3

[0332] P=1001, and

[0333] RA=X^(p+ra)/P=X³⁺¹/P, so that p+ra is greater than or equal to the degree of each message segment M_(s−1) and M_(s−2);

[0334] then

[0335] RA=X³⁺¹/P=10000/1001=10;

[0336] where

[0337] and

[0338] T_(0(i+1))=M_(s−1)*RA=1011*10=10110,

[0339] T_(0(i))=M_(s−2)*RA=1000*10=10000,

[0340] T_(1(i+1))=T_(0(i+1))/X³⁺¹=10110/10000=1,

[0341] T_(1(i))=T_(0(i))/X³⁺¹=10000/10000=1,

[0342] T_(2(i+1))=T_(1(i+1))*P=1*1001=1001,

[0343] T_(2(i))=T_(1(i))*P=1*1001=1001,

[0344] R_(i+1)=M_(s−1)−T_(2(i+1))=1011−1001=010

[0345] R_(i)=M_(s−2)−T_(2(i))=1000−1001=001.

[0346] Hence, modulo units 72 a and 72 b obtain the same remainders R_(i+1) and R_(i) as modulo units 32 a and 32 b in FIG. 3 above.

[0347] Multiplier 34 multiplies R_(i+1) and R_(i) by segment-constants C_(i+1) and C_(i) to obtain segment-remainders SR_(i+1) and SR_(i). Here, segment-constants C_(i+1) and C_(i) are obtained ‘on the fly’ by moduloing the position X of segments 33 and 35 in message M by generator polynomial P (i.e., C_(i+1)=X^(n*(i+1))(modulo)P and C_(i)=X^(n*i)(modulo)P) using modulo unit 60 described in FIG. 6. For example:

[0348] if

[0349] X^(n*(i+1))=X^(4*(1))=M₁=10000,

[0350] X^(n*i)=X^(4*(0))=M₀=00001,

[0351] Deg(X^(4*(1)))=4,

[0352] Deg(X^(4*(0)))=0,

[0353] P=1001, and

[0354] RA=X^(p+ra)/P=X³⁺¹/P, so that p+ra is greater than or equal to the degree of each message segment X^(4*(1)) and X^(4*(0));

[0355] then

[0356] RA=10000/1001=10;

[0357] and

[0358] T_(0(i+1))=M₁*RA =10000*10=100000,

[0359] T_(0(i))=M₀*RA=00001*10=000010,

[0360] T_(1(i+1))=T_(0(i+1))/X³⁺¹=100000/10000=10,

[0361] T_(1(i))=T_(0(i))/X³⁺¹=000010/10000=0,

[0362] T_(2(i+1))=T_(1(i+1))*P=10*1001=10010,

[0363] T_(2(i))=T_(1(i))*P=0*1001=00000,

[0364] C_(i+1)=M₁−T_(2 (i+1))=10000−10010=010,

[0365] C_(i=M) ₀−T_(2(i))=00001−00000=001.

[0366] In other embodiments segment-constants C_(i+1) and C_(i) may be obtained in advance in stored in a memory unit (e.g. 39).

[0367] Multiplier 74 includes multipliers 74 a and 74 b. Multiplier 74 a multiplies remainder R_(i+1) by segment-constant C_(i+1) to obtain segment-remainder SR_(i+1). Multiplier 74 b multiplies R_(i) by segment constant C_(i) to obtain segment-remainder SR_(i). For example:

[0368] if

[0369] R_(i+1)=010, C_(i+1)=010 and (from above)

[0370] R_(i)=001, C_(i)=001;

[0371] then

[0372] SR_(i+1)=R_(i+1)*C_(i+1)=010*010=00100, and

[0373] SR_(i)=R_(i)*C_(i)=001*001=00001;

[0374] Accumulator 76 adds segment-remainders SR_(i+1) and SR_(i) together to obtain accumulated-remainder AR. For example:

[0375] if

[0376] SR_(i+1)=SR₁=00100,

[0377] SR_(i)=SR₀=00001;

[0378] then

[0379] AR=00100+00001=00101.

[0380] Modulo unit 78 obtains a CRC for message M by moduloing accumulated-remainder AR by generator polynomial P. Here, modulo unit 78 obtains the CRC by using multiplication by reciprocal approximation shown in FIG. 6. For example:

[0381] if

[0382] AR=M=00101,

[0383] Deg(AR)=2

[0384] P=1001, and

[0385] RA=X^(p+ra)/P=³⁺¹/P so that p+ra is greater than or equal to the degree of the message for which a remainder is desired, here AR;

[0386] then

[0387] RA=10000/1001=10;

[0388] and

[0389] T₀=M*RA=00101*10=1010,

[0390] T₁=T₀/X³⁺¹=1010/10000=0,

[0391] T₂=T₁*P=0*1001=0,

[0392] R=CRC=M−T₂=00101−0=101.

[0393] Thus CRC generator 70 obtains the same CRC the example for CRC generator 30. Likewise, CRC generator 70 may also be expanded to include enough components for obtaining the CRC for message M separated into N segments.

[0394] CRC generator 80 (FIG. 8) includes subtraction unit 82, modulo unit 84 and accumulator 86 for updating a CRC of a message M adjusted during transmission. Subtraction unit 82 subtracts old message 83 from new message 85 to obtain difference D. For example:

[0395] if

[0396] then

[0397] D=M_(new)−M_(old)=00110000

[0398] wherein

[0399] Modulo unit 84 modulos difference D by generator polynomial P to obtain a difference-remainder DR. For example:

[0400] if

[0401] P=1001, and

[0402] D=00110000;

[0403] then

[0404] DR=D(modulo)P=MOD(D, P)=

[0405] wherein

[0406] In other embodiments, difference-remainder DR may be obtained using multiplication by reciprocal-approximator RA (i.e. MH (D, RA, P)).

[0407] Accumulator 86 adds difference-remainder DR and CRC_(old) to obtain a CRC_(new). For example:

[0408] if

[0409] CRC_(old)=101 and

[0410] DR=110;

[0411] then

[0412] CRC_(new)=CRC_(old)+DR=101+110

[0413] =011;

[0414] where

[0415] The accuracy of this CRC_(new) may be confirmed by replacing CRC_(old) in the adjusted message M_(new) with CRC_(new) and determining whether M_(new)(modulo)CRC_(new) equals zero. For example: $\begin{matrix} {\quad 101} \\ \underset{\_}{+ 110} \end{matrix}$   011 = CRC_(new).

[0416] CRC generator 90 (FIG. 9) includes subtraction unit 92, modulo unit 94, multiplier 96, modulo unit 98 and accumulator 99 for updating a CRC of a message M adjusted during transmission. CRC generator 90 differs from generator 80 in that it adjusts the CRC of a message M based on the adjusted segment of the message.

[0417] Subtraction unit 92 subtracts old message segment 93 from new message segment 95 to obtain difference-segment DS. For example:

[0418] if

[0419] P=1001,

[0420] n=2

[0421] s=4

[0422] M_(new)=10001|101

[0423] M_(s−1)=10

[0424] M_(s−2)=00=segment 95

[0425] M_(s−3)=11

[0426] M_(s−4)=01

[0427] then

[0428] DS=M_(s−2(new))−M_(s−2(old))=00−11=11.

[0429] Modulo unit 94 modulos difference-segment DS by generator polynomial P to obtain a difference-segment-remainder DSR. For example:

[0430] if

[0431] P=1001, and

[0432] DS=11;

[0433] then

[0434] DSR=DS(modulo)P=MOD(DS, P)=11

[0435] wherein

[0436] Here, as above, if the difference-segment DS is of a lesser degree than polynomial P, modulo unit 94 is not needed since the modulo of DS equals DS.

[0437] Multiplier 96 multiplies difference-segment-remainder DSR by an appropriate segment-constant C_(i) to obtain an expanded segment-remainder ESR. Segment-constants C₃ to C₀ for this example may be obtained as described above. For example:

[0438] if

[0439] DSR=(M_(2new−)M_(2old))(modulo)P=11

[0440] and

[0441] C_(i)=C₂=X^(2*2)(modulo)P

[0442] =10000(modulo)1001=010;

[0443] then

[0444] EDR=DSR*C_(i)=11*010=110.

[0445] Modulo unit 98 obtains message difference-remainder DR by moduloing the extended difference-remainder by generator polynomial P. For example:

[0446] if

[0447] P=1001 and

[0448] EDR=110;

[0449] then

[0450] DR=110.

[0451] Again, for extended difference-remainders of a degree less than the degree of polynomial P the DR is the EDR.

[0452] Finally, accumulator 99 adds the message difference-remainder DR and CRC_(old) to obtain a CRC_(new). For example:

[0453] if

[0454] CRC_(old)=101 and

[0455] DR=110;

[0456] then

[0457] CRC_(new)=CRC_(old)+DR=101+110

[0458] =011;

[0459] wherein $\begin{matrix} {\quad 101} & \quad & \quad \\ \underset{\_}{+ \quad 110} & \quad & \quad \\ {\quad 011} & = & {{CRC}_{new}.} \end{matrix}$

[0460] All of the above algorithms may be affected by embedding generator polynomial P in a larger ring. For example, let

F=P*Q;

[0461] where F is a field extension of P, Q is an extender, and the greatest common denominator between P and Q is one (1). Segment-constants C may now be calculated using field extension F, instead of p, and message segments M_(s) increased in size (by bit) accordingly without requiring the additional modulos 42 and 42 in FIGS. 3 and 4 above. Rather, only modulo by P, as shown in FIG. 5 may be needed.

[0462]FIG. 10 shows a general-purpose computer 100 for obtaining a CRC using process 20 or any of the operations of the CRC generator units 30, 40, 50, 60, 70, 80 and 90 shown above. Computer 100 includes a processor 102 (e.g. a CPU), a storage medium 104 (e.g., a random access memory) and communication interface 106 (e.g., a network card) having one or more external connections 106 a, 106 b and 106 c for sending and receiving data transmissions. Storage medium 104 stores computer instructions 108 for obtaining a CRC via process 20 or the operations of the CRC generator units described above. In one embodiment, computer 100 obtains a CRC for a message M based on multiplication by reciprocal approximation.

[0463] Process 20 and the operations of the CRC generators shown above, however, are not limited to use with any particular hardware or software configuration; they may find compatibility in any computing or processing environment. Process 20 may be implemented in hardware, software, or any combination of the two. So too, may the operations of the CRC generator units 30, 40, 50, 60, 70, 80 and 90.

[0464] Process 20 and the CRC generators described above may be implemented in computer programs executing on programmable computers that each include a processor, a storage medium readable by the processor (e.g. volatile memory, non-volatile memory, etc.), one or more input devices, and one or more out devices. Program code may be applied to data entered using an input device to perform process 20 or any of the operations of the CRC generators described above. The output information may be applied to one or more output devices, such as screen 110.

[0465] Each such program may be implemented in a high level procedural or object-oriented programming language to communicate with a computer system. However, the programs can be implemented in assembly or machine language. The language may be a compiled or an interpreted language.

[0466] Each computer program may be stored on an article of manufacture, such as a CD-ROM, hard disk, or magnetic diskette, that is readable by computer 100 to obtain a CRC for message M in the manners described above. Process 20 and the operations for implementing the CRC generators above may also be implemented as a machine-readable storage medium, configured with one or more computer programs, where, upon execution, instructions in the computer program(s) cause the processor 102 to operate as described above.

[0467] A number of embodiments of the invention have been described. Nevertheless, it will be understood that various modifications may be made without departing from the spirit and scope of the invention. For example, message M may be divided into an odd number of segments or segment sizes or field extensions F may be substituted for generator polynomial P were appropriate. Accordingly, other embodiments are within the scope of the following claims. 

What is claimed is:
 1. A method for obtaining a cyclic redundancy code for a message, comprising: separating the message into a plurality of segments; multiplying a remainder for each segment by a segment-constant based on a generator polynomial to obtain a plurality of segment-remainders; accumulating the segment-remainders to obtain an accumulated-remainder; and moduloing the accumulated-remainder by the generator polynomial to obtain the cyclic redundancy code for the message.
 2. The method of claim 1, further comprising, moduloing the segments by the generator polynomial to obtain the remainder for each segment.
 3. The method of claim 1, further comprising separating the message into three or more segments.
 4. The method of claim 1, wherein the cyclic redundancy code is appended to the message and the appended message is transmitted to a receiver.
 5. The method of claim 1, wherein cyclic redundancy code indicates the existence of an error in the message.
 6. The method of claim 5, wherein integrity of the message is verified if the cyclic redundancy code is zero.
 7. The method of claim 5, wherein the integrity of the message invalidated if the cyclic redundancy code is non-zero.
 8. The method of claim 1, wherein moduloing includes dividing by the generator polynomial.
 9. The method of claim 1, wherein moduloing includes multiplying by a reciprocal-approximator for the generator polynomial.
 10. The method of claim 1 wherein the segment-constant for each segment is obtained by moduloing the position of the segment in the message by the generator polynomial.
 11. A device for obtaining a cyclic redundancy code for a message, the message separated into a plurality of segments, comprising: a multiplier to multiply a remainder for each segment by a segment-constant based on a generator polynomial to obtain a plurality of segment-remainders; an accumulator to accumulate the segment-remainders to obtain an accumulated-remainder; and a modulo unit to modulo the accumulated-remainder by the generator polynomial to obtain the cyclic redundancy code for the message.
 12. The device in claim 11, wherein the device is a network card and the modulo unit includes a plurality of modulo units to modulo the each segment of the message by the generator polynomial to obtain the remainder for each segment.
 13. The device in claim 11, further comprising a memory for storing a plurality of segment-constants.
 14. The device in claim 11, wherein the segments constants obtain upon receipt of the message.
 15. The device in claim 11, wherein the modulo unit divides the accumulated-remainder by the generator polynomial to obtain the cyclic redundancy code.
 16. The device in claim 11, wherein the modulo unit multiplies the accumulated-remainder by a reciprocal-approximator for the generator polynomial to obtain the cyclic redundancy code.
 17. A method for determining a cyclic redundancy code, comprising: separating a message into a plurality of segments; multiplying each segment by a segment-constant based on a generator polynomial to obtain a plurality of segment-remainders; accumulating the segment-remainders to obtain an accumulated-remainder; and moduloing the accumulated-remainder by the generator polynomial to obtain the cyclic redundancy code for the message.
 18. The method of claim 17, where a degree of a most significant bit of the generator polynomial is greater than a degree of a most significant bit of each segment.
 19. The method of claim 17, comprising separating the message into three or more segments.
 20. The method of claim 17, wherein the generator polynomial includes a field extender.
 21. The method of claim 17, wherein cyclic redundancy code indicates a likelihood of an error in the message.
 22. The method of claim 17, wherein each one the plurality of segment-constants is based on the generator polynomial and the position of the segment in the message.
 23. A device that obtains a cyclic redundancy code for a message, the message separated into a plurality of segments, comprising: a multiplier to multiply each segment by a segment-constant to obtain a plurality of segment-remainders; an accumulator to accumulate the segment-remainders to obtain an accumulated-remainder for the message; and a modulo unit to modulo the accumulated-remainder by a generator polynomial to obtain the cyclic redundancy code for the message.
 24. The device in claim 23, further comprising a memory for storing a plurality of segment-constants.
 25. The device in claim 23, wherein the modulo unit divides the accumulated-remainder by the generator polynomial to obtain the cyclic redundancy code.
 26. The device in claim 23, wherein the modulo unit multiplies the accumulated-remainder by a reciprocal-approximator for the generator polynomial to obtain the cyclic redundancy code.
 27. A method for incrementally updating a cyclic redundancy code, comprising: subtracting a prior message from an updated message to obtain a difference; moduloing the difference by a generator polynomial to obtain a remainder; and adding the remainder to a cyclic redundancy code for the prior message to obtain an updated cyclic redundancy code for the updated message.
 28. The method in claim 27, wherein the moduloing includes multiplying the accumulated-remainder by a reciprocal-approximator for the generator polynomial to obtain the remainder.
 29. A device which incrementally updates a cyclic redundancy code, comprising: a subtraction unit to subtract a prior message from an updated message to obtain a difference; a modulo unit to modulo the difference by a generator polynomial to obtain a remainder; and an accumulator to add the remainder to a cyclic redundancy code for the prior message to obtain an updated cyclic redundancy code for the updated message.
 30. The device in claim 29, wherein the subtraction unit includes exclusive-or logic gates.
 31. A method for incrementally updating a cyclic redundancy code for a message, comprising: subtracting a prior message segment from an updated message segment to obtain a difference-segment; moduloing the difference-segment by a generator polynomial to obtain a difference segment-remainder; multiplying the difference segment-remainder by a segment-constant to obtain an expanded segment-remainder; moduloing the expanded segment-remainder by the generator polynomial to obtain an updated message-remainder; and adding the updated message-remainder to a cyclic redundancy code for the prior message to obtain an updated cyclic redundancy code for the updated message.
 32. The method in claim 31, wherein moduloing includes dividing by the generator polynomial.
 33. A device which incrementally updates a cyclic redundancy code for a message, comprising: a subtraction unit to subtract a prior message segment from an updated message segment to obtain a difference-segment; a modulo unit to modulo the difference-segment by a polynomial to obtain a difference segment-remainder; a multiplier to multiply the difference segment-remainder by a segment-constant to obtain an expanded segment-remainder; a modulo unit to modulo the expanded segment-remainder by the polynomial to obtain an difference-remainder; and an accumulator to add the difference-remainder to a prior cyclic redundancy code for the prior message to obtain an updated cyclic redundancy code for the updated message.
 34. The device in claim 29, wherein the accumulator includes exclusive-or logic gates.
 35. An article comprising a machine-readable medium that stores instructions to obtain a cyclic redundancy code for a message, the instructions causing a machine to: separate the message into a plurality of segments; multiply a remainder for each segment by a segment-constant based on a generator polynomial to obtain a plurality of segment-remainders; accumulate the segment-remainders to obtain an accumulated-remainder; and modulo the accumulated-remainder by the generator polynomial to obtain the cyclic redundancy code for the message.
 36. The article of claim 35, further comprising instructions that cause a machine to modulo the segments by the generator polynomial to obtain the remainder for each segment.
 37. The article of claim 35, further comprising instructions that cause a machine to verify the integrity of the message if the cyclic redundancy code is zero.
 38. The article of claim 35, further comprising instructions that cause a machine to invalidate the integrity of the message if the cyclic redundancy code is non-zero.
 39. An article comprising a machine-readable medium that stores instructions to obtain a cyclic redundancy code for a message, the instructions causing a machine to: separate a message into a plurality of segments; multiply each segment by a segment-constant based on a generator polynomial to obtain a plurality of segment-remainders; accumulate the segment-remainders to obtain an accumulated-remainder; and modulo the accumulated-remainder by the generator polynomial to obtain the cyclic redundancy code for the message.
 40. The article of claim 39, further comprising instructions that cause a machine to apply a field extender to the generator polynomial.
 41. An article comprising a machine-readable medium that stores instructions to obtain a cyclic redundancy code for a message, the instructions causing a machine to: subtract a prior message from an updated message to obtain a difference; modulo the difference by a generator polynomial to obtain a remainder; and add the remainder to a cyclic redundancy code for the prior message to obtain an updated cyclic redundancy code for the updated message.
 42. The article of claim 41, further comprising instructions that cause a machine to obtain the remainder by multiplying the accumulated-remainder by a reciprocal-approximator for the generator polynomial.
 43. An article comprising a machine-readable medium that stores instructions to obtain a cyclic redundancy code for a message, the instructions causing a machine to: subtract a prior message segment from an updated message segment to obtain a difference-segment; modulo the difference-segment by a generator polynomial to obtain a difference segment-remainder; multiply the difference segment-remainder by a segment-constant to obtain an expanded segment-remainder; modulo the expanded segment-remainder by the generator polynomial to obtain an updated message-remainder; and add the updated message-remainder to a cyclic redundancy code for the prior message to obtain an updated cyclic redundancy code for the updated message.
 44. The article of claim 43, further comprising instructions that cause a machine to modulo by division.
 45. The article of claim 43, further comprising instructions that cause a machine to modulo by reciprocal approximation. 